Çözümlü Problemler / Solved Problems

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From the course by Koç University

Doğrusal Cebir II: Kare Matrisler, Hesaplama Yöntemleri ve Uygulamalar / Linear Algebra II: Square Matrices, Calculation Methods and Applications

13 оценки

Koç University

Doğrusal Cebir II: Kare Matrisler, Hesaplama Yöntemleri ve Uygulamalar / Linear Algebra II: Square Matrices, Calculation Methods and Applications

13 оценки

Doğrusal cebir ikili dizinin ikincisi olan bu ders birinci derste verilen temel bilgilerin üzerine eklemeler yapılarak tamamen matris işlemleri ve uygulamalarını kapsamaktadır. Cebirsel denklem sistemleri, sonuçların tekilliği ve var olup olmadığı, determinantlar ve onların doğal olarak nasıl oluştuğu, öz değer problemleri ve onların matris fonksiyonlarına uygulanışı vb. konulara derste değinilmektedir. Ders gerçek yaşamdan gelen uygulamaları da tanıtmaya önem veren “içerikli yaklaşımla” tasarlanmıştır. Bölümler: Bölüm 1: Doğrusal Cebir I'in Özeti Bölüm 2: Kare Matrislerde Determinant Bölüm 3: Kare Matrislerin Tersi Bölüm 4: Kare Matrislerde Özdeğer Sorunu Bölüm 5: Matrislerin Köşegenleştirilmesi Bölüm 6: Matris Fonksiyonları Bölüm 7: Matrislerle Diferansiyel Denklem Takımları ----------- This second of the sequence of two courses builds on the fundamentals of the first course, is entirely on matrix algebra and applications. Specifically, the studies include systems of algebraic equations including the existence and uniqueness of solutions, determinants and how they arise naturally, eigenvalue problems with their applications to diagonalization and matrix functions. The course is designed in the same spirit as the first one with a “content based” emphasis, answering the “why” and “where“ of the topics, as much as the traditional “what” and “how” leading to “definitions” and “proofs”. Chapters: Chapter 1: Summary of Linear Algebra I Chapter 2: Determinant Chapter 3: Inverse of Square Matrices Chapter 4: Eigenvalue Problem in Square Matrices Chapter 5: Diagonalization of Matrices Chapter 6: Matrix Functions Chapter 7: Matrices and Systems of Differential Equations ----------- Kaynak: Attila Aşkar, “Doğrusal cebir”. Bu kitap dört ciltlik dizinin üçüncü cildidir. Dizinin diğer kitapları Cilt 1 “Tek değişkenli fonksiyonlarda türev ve entegral”, Cilt 2: "Çok değişkenli fonksiyonlarda türev ve entegral" ve Cilt 4: “Diferansiyel denklemler” dir. Source: Attila Aşkar, Linear Algebra, Volume 3 of the set of Vol1: Calculus of Single Variable Functions, Volume 2: Calculus of Multivariable Functions and Volume 4: Differential Equations.

From the lesson

Matrislerin Köşegenleştirilmesi / Diagonalization of Matrices

Matrislerin Köşegenleştirilmesi / Diagonalization of Matrices9:00

Çözümlü Problemler / Solved Problems10:42

Simetrik Matrislerin Her Koşulda Köşegenleştirilebilmesi / Diagonalization of Symmetric Matrices14:00

Matris Dönüşümleri / Matrix Transformations27:17

Matrislerin Değişmezleri ve İşareti Tanımlı Matrisler / Invariants of Matrices and Definite Matrices20:12

Simetrik Matrisler ve İkinci Derece Kuvvet Fonksiyonları / Symmetric Matrices and Quadratic Functions26:26

Meet the Instructors

Attila Aşkar
Prof. Dr.
Matematik Bölümü (Department of Mathematics)

[BOŞ_SES] Hello,

Diagonalization of martisor We have seen the subject.

As we were doing; Matrix given us, a full matrix.

We find that the self-vector, we use as columns.

We call it the Q matrix.

We calculate the inverse of this matrix Q, we left them stand out.

This product, the triple product gives us a diagonal matrix.

Now we can say this is why it's important.

A correlation matrix table.

For example, the number one event in two A1 2

Or, number two is a relationship off the entry.

And such that,

It indicates that each event is associated with other full matrix.

But when you make a diagonal, this one consisting of the event

Each of the series of complex events from an event

reducing the possibility of the occurrence of the event giving one event.

This is a great achievement but it's because we're all intelligent people

a mixed event taking grains difficult to understand,

but it could reduce the grain into an independent event,

We understand supremely easy.

Here comes the achievement of power diagonalization.Lines here.

Now let's move to our example.

Examples will still numerically simple examples.

One, two, three, four matrices have also been studied previously already.

We want to diagonalization of this matrix.

Our work, find eigenvalues of this matrix.

We write the matrix, one, two, three, four.

Minus minus lambda lambda are writing on the diagonal.

This eigenfunction of, we calculate the characteristic function.

There is here a lambda lambda squared minus,

There's less here in four lamps, five lamps,

here we find the four minus six.

That we have already found the root of minus one and five before.

These corresponding

We also found that the extract vector minus one and a two.

You can find it by looking for more details of the previous examples.

The two independent vectors.

We already had the theorem about it.

If it come opposed to a discrete value independent eigenvectors.

I do now, but they are not steep.

Because of this, we see that the matrix be symmetrical upright.

Q matrix to create the work we do.

extract the first column vector e1, e2 are writing the core vector in the second column.

Q vector output.

We find the inverse of this Q vector.

We found the opposite, is not hard to find.

Here we find the cofactor it.

And discarding the first line of the first column, two coming, we wrote two on top.

etc. The opposite of these things,

We calculate the inverse of the two binary matrix.

Determinantına da böleceğiz.

Determinant will be three turns.

Here we find the Q-1.

This is multiplied by the time we left Q-1 to Q

We find the numbers on the right and diagonal minus one to five

minus one because we chose as the first core value, we chose the second five.

Vice versa if we had, then it would consist of five and minus on a diagonal.

No need to make this account.

Theorem gives us the guarantee.

In fact, you do not need to calculate the Q-1.

Q. We know the times, we know what it is, we multiply with Q-1,

theorem gives us that out of this diagonal matrix.

But just to see concretely how this occurs,

Let us write out the AI.

Q right, let's write the Q-1 to the left.

We can easily see that these opposite one another.

Q-1 to Q to multiply, see also have received the first column, we turn horizontally.

Two plus three.

Three also divided.

Output.

So we're going to the unit matrix.

If we multiply the first column of the second line,

See minus two plus two; zero, we divide by three.

That is to say, one and zero output.

Again takes the second column

If we multiply the first line, we see that a plus minus an avalanche.

Ie zero three de bölsek still zero.

Minus one divided by three, divided by the time we hit a three in the second row, plus one.

Plus also comes with a three-three split we split three to two.

So on a diagonal, diagonal outside zero

We find that when we hit Q'yl Q-1.

Similarly, if we are putting forth this product,

I did not do it, you can be an easy exercise.

Really seen as a negative on the diagonal and five out.

Let's take a slightly larger matrix.

This matrix of self-worth and self

In the example we made page one hundred forty nine vectors.

We still do a previous example in this respect, self-worth and self-vectors.

We found two four six.

By contrast, we found that self-vectors like that.

This eigenvectors

by type in columns, we find the Q vector Q matrix.

Q matrix on the right, the left, if we multiply by Q reverse,

theorem tells us that it would be a diagonal matrix.

It slams this as shown below, we can prove it.

But you do not need to make this product.

So if you want to ensure there is a benefit in any way.

Theorem it guarantees.

Accounts fig we do it right, to do so for us it's theorem,

This product will give the diagonal matrix and will give this order,

It was opposing the smallest eigenvalues e1,

two, four, and this is from six opponents were self-vector.

Now, as another example, however, a matrix of three to three.

Yet we also solved it.

153.

In the example we solve this page.

There was a special case here.

Lambda one; Lambda Lambda three together two and five but was out evenly.

But even if repeated eigenvalues,

We found two lambda and lambda in the E2 and E3 self-vector corresponding to three.

These vectors are independent from each other.

Lambda also independent from the opposite vector is already here

As it has others.

Theorem says that; independent self-worth,

eigenvectors are independent from each other from opposite to different core values.

Lambda from an opposite e1, e2 and e2 and the e3 independent but

e3 is independent but there was no guarantee that they are independent in this case turns out.

Remember that if you can find examples of self-worth,

If you are having trouble finding eigenvectors.

Yet this self-vectors of position the columns, the first column,

For the second column and the third column, we find the Q matrix.

Left to the inverse of the matrix Q

If we hit the right A'yl including himself,

The theorem gives us the guarantee that it will be a diagonal matrix.

And also on the opposing diagonal e1 five core values,

e2 and e3 opposite

arrivals were from the same core values as vectors.

It also gives this number are repeated on the diagonal.

We had another example again in the same context,

In the example hundred fifty sixth page.

Here a triangular matrix to facilitate the sheer account.

See also this core value of five, one, we see that one.

But here found two eigenvectors against repeated nineteen core values.

Thus, the matrix can not be diagonalized.

So this matrix, for example, has given us all

the show is going to be köşegenleştirileme matrix.

These examples're staying here.

We will then move to symmetric matrices.

If you remember, you also have to remember that in order to advance.

There were strengths in the core values of the symmetric matrix problems.

We do this in the next session.

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